When Kloosterman sums meet Hecke eigenvalues

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from $\mathrm{\ell }$-adic cohomology, as well as a Bombieri–Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke–Maass cusp form of trivial nebentypus, the eigenvalue of the $n$-th Hecke operator does not coincide with the Kloosterman sum $\mathrm{K}\mathrm{l}(1,n)$ for infinitely many squarefree $n$ with at most $100$ prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterman sums.